Motion planning in robotics requires not only computing collision-free paths but also certifying infeasibility when no such path exists. Complete methods are limited to low-dimensional spaces, while sampling-based planners scale efficiently but cannot provide finite-time infeasibility certificates, leaving this problem largely unresolved in high-dimensional spaces. In this letter, we present a geometry-driven framework for certifying infeasibility through an explicit resolution-dependent analysis of configuration space topology. Leveraging signed distance field representations, the proposed method traces separating manifolds induced by obstacle boundaries directly in configuration space, enabling both detection of infeasibility and identification of the specific geometric cause. To address computational challenges, we develop a parallel frontier-expansion algorithm that exploits GPU acceleration for efficient simplicial reconstruction in high-dimensional spaces. We validate the approach on 4-DOF and 5-DOF robot scenarios, certifying infeasibility within seconds for 4-DOF cases and under four minutes for 5-DOF cases. We further discuss avenues for improving scalability to higher-dimensional spaces.

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